Consider a matrix $\mathrm { A } = \left[ \begin{array} { c c c } \alpha & \beta & \gamma \\ \alpha ^ { 2 } & \beta ^ { 2 } & \gamma ^ { 2 } \\ \beta + \gamma & \gamma + \alpha & \alpha + \beta \end{array} \right]$, where $\alpha , \beta , \gamma$ are three distinct natural numbers. If $\frac { \operatorname { det } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } ( \operatorname { adj } A ) ) ) } { ( \alpha - \beta ) ^ { 16 } ( \beta - \gamma ) ^ { 16 } ( \gamma - \alpha ) ^ { 16 } } = 2 ^ { 32 } \times 3 ^ { 16 }$, then the number of such 3-tuples $( \alpha , \beta , \gamma )$ is $\_\_\_\_$ .